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Vague the question: how long have all the mathematicians working in the respective fields known the theory of categories?

More specific questions:

  1. Is it true that all modern working algebraic topologists know category theory? How long ago?
  2. Is it true that all modern working algebraists know category theory? How long ago?

Questions still remain somewhat vague. Of course, I am not interested in single exceptions. I got the impression that in many areas of modern mathematics the language of category theory (category, functor, limit, colimit, adjoint functor, etc.) is widely used in articles and therefore it is impossible to study this area without knowing the theory of categories. I'm wondering for which areas this is really true and for how long ago.

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    $\begingroup$ "All" do not "know" it in the sense of using it even today, there are large parts of algebra and topology that do not really need it either. But it became widespread and "heard of" in algebraic topology since Eilenberg and MacLane axiomatized homology in 1940s, and in algebra/algebraic geometry since Grothendieck and Bourbaki started using topoi, etc., in 1950s. MacLane's 1971 book helped spread it into the masses beyond the cutting edge researchers. $\endgroup$
    – Conifold
    Commented Nov 19, 2021 at 12:08
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    $\begingroup$ Many group-theorists (that I know) do not use the categorical language and even dislike it. The same applies to, say, researchers working in non-associative rings. On the other hand, I am yet to meet an algebraic geometer who did not use the categorical language. $\endgroup$
    – Moishe Kohan
    Commented Nov 19, 2021 at 13:53
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    $\begingroup$ For quite a few people, including myself, basic ideas of category theory are explanatory and help organizing things, much as basic ideas of set theory help in certain ways. I do not find myself invoking subtle theorems from either category theory or set theory, but just using basic concepts as descriptive, explanatory, organizational devices. Not as prerequisites. $\endgroup$
    – paul garrett
    Commented Nov 19, 2021 at 17:16
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    $\begingroup$ Somewhat related: Should every modern day mathematician care about category theory? (3 answers and many relevant comments) $\endgroup$
    – Dave L Renfro
    Commented Nov 20, 2021 at 12:41

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I am an algebraist. In the first edition of Lang's "Algebra", category theory was called "abstract nonsense". I bought the book 47 years ago. That is how long I know it. It is still abstract nonsense as far as I know. It is quite possible to study my part of algebra (geometric and algorithmic group theory) without knowing any category theory.

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  • $\begingroup$ Same book, bought 44 years ago . . . $\endgroup$
    – Dave L Renfro
    Commented Nov 20, 2021 at 12:53
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    $\begingroup$ Yes, perhaps oddly, the early editions of his "Algebra" seemed to treat categorical stuff very dismissively, and homological stuff even more so: "take any book on homological algebra and prove all the theorems in it without looking at the proofs in that book". Of course, (from my direct acquaintance with him long ago), Lang was never one to hesitate or be shy. :) $\endgroup$
    – paul garrett
    Commented Nov 20, 2021 at 19:08

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